TRANSACTIONS OF SOCIETY OF ACTUARIES
1 9 9 0 VOL. 4 2
THE IMPACT OF MORTALITY ON PANJER'S MODEL
OF AIDS SURVIVAL
COLIN M. RAMSAY
ABSTRACT
This paper is intended to be a sequel to Panjer's paper [12]. Using Ramsay's [13] extension to Panjer's model, we prove that Panjer's results are
biased. In particular, his estimates of the transition intensities are too high
because he did not include the mortality risk that exists before a life develops
full-blown AIDS. Panjer thus understates the average time that it will take
an HIV+ life to develop AIDS or to die of AIDS. Suggestions are made
for gathering data in an AIDS environment.
1. I N T R O D U C T I O N
The Walter Reed Staging Method (WRSM), as described by Redfield et
al. [14], is a method that groups patients infected with the Human Immunodeficiency Virus (HIV) into various stages, the final stage being AIDS.
The WRSM was used by Panjer [12] to describe the progression of the
disease process once a life became infected with HIV. His model essentially
says that the progression of the disease through its various stages is sequential
and irreversible. The stages are:
Stage 0 (At-risk) Healthy persons at risk for HIV infection, but testing negative
Stage 1 (HIV+) Otherwiseasymptomatiepersons testing HIV +
Stage 2 (LAS) Personswith HIV infectionand lymphademopathysyndrome(I..AS),
together with moderate celluar immune deficiency
Stage 3 (ARC) Patients with HIV infection and LAS, togetherwith severe cellular
immune deficiency(AIDS-RelatedComplex, or ARC)
Stage 4 (AIDS) Patientswith acquired immune deficiencysyndrome(AIDS)
Stage 5 (Death) Patientswho died "of AIDS.'"
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Stage 6 (Death) Personswho died in stages 0, 1, 2, or 3. Persons who died in stage
4 of causes not related to AIDS.
In stage 0 are healthy lives that are not currently testing positive for HIV
but have a relatively high probability of becoming infected with the virus.
Such a group of lives is called an "at-risk" group. Several factors contribute
to exposing a life to the risk of becoming HIV + . The factors cited most
often are IV drug use, sexual orientation and sexual promiscuity.
321
322
IMPACT OF MORTALITY ON PANJER'S MODEL
To avoid confusion, we must define the conditions under which the term
" A I D S " is used. In this paper a life is said to have AIDS if and only if it
is in stage 4. This definition is in contrast to the popular usage that refers
to any life in stages 1 through 3 as having "AIDS." Obviously in these
earlier stages the lives have not yet developed AIDS in the sense of the
WRSM, so the term "AIDS" is not used to describe such lives. As a result,
the expression "a life died of AIDS" means that the life died in stage 4
from a disease or condition that is directly related to the AIDS condition.
Clearly a life can die at any time and in any of the stages. This fact was
not included in Panjer's model. If one assumes that the mortality pattern
exhibited by the members of an at-risk group is similar to standard mortality,
one may be able to argue that the force of mortality will be negligible when
compared to the force of transition to the next stage according to the WRSM.
However, if one thinks that lives in the high-risk groups have significantly
substandard mortality experiences, it will be necessary to include a mortality
component in the model at stage 0. Once a person has entered stage 1, it is
clear that the person's overall mortality and morbidity rates will be worse
than those of healthy uninfected lives, especially as the life progresses through
the various stages to AIDS. This is because HIV attacks the body's T4
lymphocyte cells, thus destabilizing the body's immune system and leaving
the body open to opportunistic infections. Hence in stages 1 through 3 the
death rates from life-threatening diseases will be expected to be significantly
higher in HIV + populations than in uninfected ones. In view of this, the
effects of mortality on lives before they have developed AIDS cannot be
ignored.
The inclusion of the possibility of death in any stage and the separation
of the deaths in stage 4 into AIDS- and non-AIDS-related causes are not
new ideas. This was the approach taken by the Institute of Actuaries AIDS
Working Party in its reports (see Daykin [3, pp. 142-144]) and by most
statisticians and epidemiologists developing mathematical models of the dynamics of HIV/AIDS (see, for example, Dietz [4], Hyman and Stanley [7],
Isham [8], and May [11]). In view of this, we are adding a seventh stage,
labeled stage 6, to accommodate the earlier deaths. Hence death in any of
stages 0 through 3 means immediate transition to stage 6. In stage 4 the
term "death" is reserved for death due to causes that are unrelated to AIDS
and its complications; these deaths imply immediate transition to stage 6. A
person who has died of AIDS or its complications is said to have "progressed" to stage 5.
IMPACT OF MORTALITY ON PANJER'S MODEL
323
2. THE MODEL WITH MORTALITY
Not much work has been done in the area of mathematically modeling
the transition process of HIV by using the WRSM. Simple models of the
transition dynamics of HIV have been developed by Cowell and Hosldns
[2], Panjer [12], and Ramsay [13]. Cowell and Hoskins used an essentially
nonparametric, discrete time model of the probabilities of progression. They
also included "duration since progression" as a factor in the transition probabilities. However, as Panjer pointed out, their results are highly sensitive
to their data. On the other hand, Panjer developed a model of an "at-risk
life" in stage 0 and its progression through to AIDS and then death. Death
prior to the development of full-blown AIDS was not permitted in this model.
He used a parametric, continuous time Markov process model with constant
forces of transition from one stage to the next. Ramsay extended Panjer's
model to include a death term. He then developed expressions for life insurance functions.
Following Ramsay, we adapt Panjer's model to include a death component
as follows: A person in stage i (i=0, 1, 2, 3, 4) is assumed to be subject
to a constant force of progression (V,i) out of stage i and into stage i + 1 ,
and a constant force of mortality (V.~) out of stage i and directly into stage
6. These forces are assumed to be operating simultaneously on each person.
In view of this, one must clearly distinguish between these forces for patients
with AIDS because they can die from AIDS-related diseases or from nonAiDS-related diseases. In stage 4, the former type of death is considered to
be due to ~4, while the latter is due to tx~.
The assumption of constant forces in each stage may appear to be somewhat unrealistic. However, this assumption is used quite often when developing mathematical models of HIV/AIDS transmissions (see, for example,
Dietz [4], Hyman and Stanley [7], Isham [9], and May [11]). Panjer's results
showed that this is a plausable first approximation to the actual transition
process. In keeping with the famous Occam's Razor--hypotheses should not
be complicated unnecessarily--we made the simplest and least complicated
extension to Panjer's model.
Because the forces are assumed to be constant, a "'memoryless" property
will exist. This means that the length of time already spent in the current
stage has no effect on the future length of time that the person will remain
in this stage. This fact will allow us to speak in terms of the future time
spent in a stage without having to condition on the amount of time already
spent in the stage.
324
IMPACT OF MORTALITY ON PANJER'S MODEL
Definition 1. Let Ti be the (future) time spent in stage i before entering stage
i + 1, and T~ be the (future) time spent in stage i before immediate transition
to stage 6, that is, death in stage i. Due to the "'memoryless" property
described above, the random variables {To. . . . . Ti-~, T~ are mutually independent as are the random variables.
Let f~(t) and gi(t) be the probability density functions (pdf's) of Ti and
T;, respectively; then fi(t) and gi(t) are given by
where ~ is the force of transition out of stage i, but not necessarily into
stage i + 1, that is,
~. = /z, + /~'.
(3)
At this point note that, technically, neither f~(t) nor g~(t) is a pdf because
neither of them integrates to 1. As a consequence, one cannot regard T~ and
T; as completely specified random variables because some part of their "randomness" is missing. In view of this, the following extensions will be made
to the definition of Te and TI: (1) if the life progresses to stage i + 1, then
we say that T; = ®, and (2) if the life dies while in stage i, then T~= ®. As
a result of the extended definition of T~ and T;, their moments will not exist;
that is, they will be infinite. However, the life expectancy of a life in stage
i can still be evaluated because, among other things, death (from any cause)
is certain, so random variables such as the total future lifetime until death
(from any cause) for a life currently in stage i will be well defined with
finite moments.
3. THE BIAS 1N PANJER'S ESTIMATES
Because Panjer's model did not include a mortality component in stages
0 through 3, it can be viewed as one based on lives that are destined (or
predetermined) to die of AIDS. His model is called a "conditional model"
because it is equivalent to conditioning on the event that all the lives will
eventually die of AIDS. In the conditional model the random variable T;
will obviously not exist because such deaths are not permitted.
IMPACT OF MORTALITY ON PANJER'S MODEL
325
Let f~(t) be the pdf of the random variable T~ conditioned on the event
that the life died of AIDS. For notational convenience, define the event st/
as
s~ = {the life currently in stage i will die of AIDS}
for i = 0 , 1, 2, 3, 4. Now ~(t) can be found by using the definition of
conditional probability as applied to the WRSM.
g (t)dt = Pr[t < T~ < t + dtlst, ]
Pr[t < T, <_ t + dt O sit]
Pr[sti]
Pr[t < T~ < t + dt] Vr[~,+x]
Pr[~]
By the definition of si s and Equation (1),
P
Pr[si,] =
J f~(t) Pr[si,÷l ] dt
o
= tx~ Pr[~i.l].
Therefore, the pdf of the random variable {T~I.~.}is
fl~(t) = ~(t) Pr[sl,+l])/ (/~'
~ Pr[.si~+l ])
= oei e -°~,
(4)
an exponential distribution with parameter ~.
The exponential distribution with parameter i~ was the pdf of Panjer's
version of T~; see Panjer's Equation (3). Therefore, Panjer's version of T~ is
actually the random variable {T~l.st~}.This means that Panjer in fact estimated
as instead of ~, making his estimates of the ~ ' s biased; he is overstating
the size of each p~. As a consequence, the expected time that it will take an
HIV + life to develop AIDS will be longer than the expected time given by
Panjer. In fact, in a strictly mathematical sense, this expected time will not
even exist. Techniques for deriving maximum likelihood estimates of the
parameters ~, and ~ are provided in the Appendix.
326
IMPACT OF MORTALITY ON PANJER'S MODEL
4. SUMMARY AND COMMENTS
Panjer's parametric survival model for HIV+ lives was extended to inelude mortality prior to the development of full-blown AIDS. It was proved
that Panjer's model is a conditional model with an intensity function in stage
i equal to c,i, the sum of the forces of mortality and progression to stage
i + 1. This produced a bias in Panjer's estimates of the ~i's, leading to an
overestimation of the 0.i's. This meant that the expected time it takes an
HIV + to develop AIDS will be longer than the expected time given by
Panjer. To estimate the force of progression without using mortality, one
must observe the lives in the study without the effects of mortality before
stage 4. Unfortunately, this is not possible. As a result, the questions posed
by Panjer--"What is the probability that an H I V + person will develop
AIDS within the next 3 years?" or " O f 1000 H I V + persons, how many
can we expect to have AIDS or have died of AIDS within the next 5 years?"-cannot be answered with his model. However, by using Ramsay's model
and his expressions for the various transition probabilities, these probabilities
can now easily be found.
One of the most important ingredients in the survival analysis of HIV +
lives is the data. The data used by Panjer did not contain the information
necessary to jointly estimate the mortality and transition parameters for even
the simple model described in this paper. Clearly, if we want to introduce
a more complex model, for example, one that assumes the intensity functions
are dependent on the duration since entering a stage, more sophisticated
studies and systems of data collection will be necessary. Future studies based
on the WRSM must, at a minimum, generate the following five pieces of
information to a tolerable degree of accuracy: (1) date of initial infection,
(2) date of transition to each stage, (3) date of death in study or date of exit
from study, (4) stage in which death occurs, and (5) cause of death, that is,
AIDS-related or non-AIDS-related cause of death.
The fifth piece of information is most likely to be found on a death
certificate. This immediately raises two concerns: (i) What constitutes an
AIDS-related death? and (ii) Are such deaths reliably reported? The former
concern can be dealt with by using the Centers for Disease Control's (CDC)
definition of AIDS. The CDC [15] defined AIDS in terms of a set of "indicator" diseases; deaths that result from any of these diseases are termed
AiDS-related deaths. As for the second concern, it may seem reasonable to
expect the death certificates of HIV/AIDS patients to be frequently unreliable
because attendant physicians may be reluctant to disclose the true cause of
IMPACT OF MORTALITY ON PANJER'S MODEL
327
death in such cases. However, the recent work by Hardy et al. [6] showed
that this may not be the case. After reviewing death certificates in four U.S.
cities--Washington, D.C., New York City, Boston, and Chicago--to evaluate AIDS surveillance effectiveness, Hardy et al. found nearly complete
AIDS case reporting. The level of reporting was much higher than reporting
for many other communicable diseases. For insurance purposes, the AIDS
Task Force [17, chapter 4, p. 11] suggested that a working definition of a
"suspected AIDS" death needs to be developed to help offset the veiled
and misstated cause-of-death problem. They provided 11 AIDS-related diseases that may be helpful in developing such a definition.
In closing, the number of persons developing full-blown AIDS is, metaphorically speaking, just the tip of the iceberg. Attention must be directed
to the totality of HIV-infected lives in the population. Rolland [16, p. 6]
wrote "Our enemy is HIV infection, not AIDS per se. Given that a very
high percentage of those who become infected will die, one's focus must
be on all who are infected rather than on that small subset who have progressed all the way to AIDS."
It is hoped that studies on the mortality, morbidity and transition processes
of lives in the various stages along the progression to AIDS will be undertaken, and that data will be gathered in a manner that will facilitate a parametic analysis of these processes.
REFERENCES
1. BURDEN,R.L. AND FAIRES, J.D. Numerical Analysis, 3rd ed. Boston: Pringle,
Weber & Schmidt, 1985.
2. COWELL,M.J. ANDHOSFdNS,W.H. "AIDS, HIV Mortalityand Life Insurance,"
Chapter 3 in The Impact of Aids on Life and Health Insurance Companies: A Guide
for Practicing Actuaries, Report of the Society of Actuaries Task Force on AIDS.
In TSA XL, Part II (1988): 909-72.
3. DAYraN,C.D. "A United Kingdom Perspective." In Insurance and the AIDS Epidemic, Proceedings of a Two Day Symposium. Schaumburg, I11.: Society of Actuaries, 1988, pp. 133-167.
4. DIET'Z,K. "On the Transmission Dynamicsof AIDS," Mathematical Biosciences
90 (1988): 397-414.
5. ELANDT-JOHNSON,R.C. AND JOHNSON, N.L. Survival Models and Data Analysis.
New York: Wiley, 1980.
6. HARDY,A.M., STARCHER,E.T., ET AL. "Review of Death Certificatesto Assess
Completeness of AIDS Case Reporting," Public Health Report 102 (1987): 38691.
328
IMPACT OF MORTALITY ON PANJER'S MODEL
7. HYMAN, J.M. AND STANLEY, E.A. "Using Mathematical Models to Understand
the AIDS Epidemic," Mathematical Biosciences 90 (1988): 415-73.
8. IMAGAWA, D.T., ET AL. "Human Immunodeficiency Virus Type 1 Infection in
Homosexual Men Who Remain Seronegative for Prolonged Periods," New England
Journal of Medicine 320, no. 22 (June 1, 1989): 1458-62.
9. ISHAM,V. "'Mathematical Modelling of the Transmission Dynamics of HIV Infection and AIDS: A Review," Journal of the Royal Statistical Society Set. A, Part 1
(1988): 5-30.
10. KENDALL,SIR M.G. AND STUART,A. The Advanced Theory of Statistics, Vol. 2,
4th ed. New York: Macmillan, 1979.
11. MAY, R.M., ANDERSON, R.M AND McLEAN, A.R. "Possible Demographic Consequences of HIV/AIDS Epidemics. 1. Assuming HIV Infection Always Leads to
AIDS," Mathematical Biosciences 90 (1988): 475-505.
12. PANJER, H.H. "AIDS: Survival Analysis of Persons Testing H W + , " TSA XL
(1988): 517-30.
13. IL~MSAV,C.M. "'AIDS and the Calculation of Life Insurance Functions," TSA XLI
(1989): 393-422.
14. REDF/ELD,R., WRIGHT,D.C. AND TgaMOt,rr, E.C. "The Walter Reed Staging
Classification for HIV-III / LAV Infection," New England Journal of Medicine 314,
no. 2 (1986): 131-32.
15. "Revision of the CDC Surveillance Case Definition for Acquired Immunodeficiency
Syndrome," Centers for Disease Control Morbidity and Mortality Weekly Supplement 36, no. 1S (1987).
16. ROLL.AND,I.M. "The Impact of AIDS on American and Canadian Insurers." In
Insurance and the AIDS Epidemic, Proceedings of a Two Day Symposium. Schaumburg, I11.: Society of Actuaries, 1988, pp. 1-21.
17. SOCIETYOF ACTUARIESAIDS TASKFORCE. "The Impact of AIDS on Life and
Health Insurance Companies: A Guide for Practicing Actuaries," TSA XL, Part II
(1988): 839-1160.
APPENDIX
ESTIMATION OF PARAMETERS
Let us consider a study consisting of m observation periods, thej-th period
being (aj, bj),j = 1, 2 . . . . , m. A group of lives n,j in some stage arbitrarily
labelled i is followed from the start of the j-th period until the end of this
period. During this period lives can remain in stage i, progress to stage i + 1
and beyond, or die while in stage i. The number of persons dying while in
stage i and the number of persons progressing to stage i + 1 and beyond are
recorded separately. The information gathered from each of the m periods
will be used to estimate the forces of mortality and progression.
IMPACT OF MORTALITY ON PANJER'S MODEL
329
NOTATION
For i = 0 , 1, 2, 3, 4 and j = l ,
. . . , m, define
/z, = force of progression from stage i to i + 1
/x" = force of progression from stage i to 6
+
d;i = number of lives progressing from stage i to i + 1 during observation
periodj
d,~. = number of lives progressing from stage i to stage 6 during observation period j
Po = probability that a life in stage i at the start of period j will remain in
stage i throughout this period
qu = probability that a life in stage i at the start of periodj will enter stage
i + 1 during period j
~. = probability that a life in stage i will progress directly to stage 6.
Because of the "memoryless" property, the probabilities defined above will
depend only on the length of period j and the current stage i. The subscript
i will be dropped from all symbols. Clearly the probabilities introduced
above are given by
~
fij =
L
bj-ai)
ore- '~ dt
= e -'~bj-'j)
(5)
ftbj-,l) Iae-'~t dt
~/J ~'~- dO
= /x
(1 - e-,~t,,-aj))
cg
(6)
f (bj-aj)
7t5 -- ,o
i~ ' e -'~ dt
= - - (1 - e -'~t'j-'j))
ot
(7)
with the subscripts i dropped.
From Elandt-Johnson and Johnson [5, chapter 12.2.1, p. 324, equation
12.4], the likelihood L is the product of m multinomial distributions, that
is,
330
IMPACT OF MORTALITY ON PANJER'S MODEL
L(/z,/,L') = fix
J-,
) qJ
- "J ?/./v/5/"J-dJ-dT~
nj
.
.
d] (nj - dy - d]> dj
while the log-likelihood is given by
l(IAlz') = const + ~ [(nj - dj - d]) logffy
j=l
+ dj log ~ + d] log ?b'].
(8)
To obtain the maximum likelihood estimators of tx and Ix', we must solve
the pair of simultaneous equations Ùl/otx = 0 and Ol/otz' = 0, where
(9)
and
o~'
~j
.,.-~
a#
Z/.,.a~'
+
(10)
From Equations (25) to (27), it is seen that
a,)p,
(11)
a~j = - ( O j - aj)pj
(12)
=
-(bj
-
Olx'
~q_~j
lz ' _
OIx = --~qJ + -~(bj - a.i)~j
O/.z
O/z
:
+
-
a,S
= ----~, ~. + -dOj - aj)pj
,u,a J
(13)
(14)
(15)
I
(16)
IMPACT OF MORTALITY ON PANJER'S MODEL
331
Substituting Equations (11) to (16) into Equations (9) and (10) will yield the
desired pair of simultaneous equations for finding the maximum likelihood
estimates of ~ and !~'. One can use the multivariate version of the NewtonRaphson method described in Burden and Faires [1, chapter 9.2] to obtain
these estimates as follows: Let
and
J(~) =
In order to solve the equation
= 0,
al,
one must choose a starting value/k ira, then for k = 0, 1, ... define
(17)
An intuitive starting value can be found by using the estimator given by
Elandt-Johnson and Johnson [5, chapter 12.2.1, p. 325, equation 12.5], that
is,
332
IMPACT OF MORTALITY
/1,(0)
=
/,i/(O)
=
1"[
dj
ON PANJER'S
I
m ,.~.~ (d; + dj)(b, - a,)
1 "[(d; dj'
-
1
MODEL
log (ni - di - d'l)
(18)
log (ni - di - d'i)
(19)
+ 4)(bj - aj)
If t2 is the maximum likelihood estimator of ~, it is well-known that 12 is
asymptotically multivariate normal with mean-p, and variance-covari~ce
matrix - [3" (/2)] -x (see, for example, Kendall and Stuart [10, chapter 18]).
In the analysis described above, it is assumed that the lives are followed
throughout the entire period, and the period did not necessarily start from
time 0. However, Panjer assumed that lives were observed over some interval (0, tj) where tj is a random variable that lies in the range (a~, ~ . Two
different scenarios were postulated in his paper: (1) lives were observed only
up to the midpoint of each range, and (2) there was a uniform censoring
mechanism for each observation. His assumptions will affect our calculations
only through their effects on ~ ~j, and ~.
1. Midpoint Assumption
Here it is assumed that for period j, tj is not random but is the midpoint
of the range. This leads to
t, = a; + b;
2
ffj =
(20)
I/ e -= dt
= e-*"
(21)
= foJlae-*~dt
= #" (1 - e-*' 0
(22)
Ot
O; =
I£j ~'e -'~ dt
= - - (1 - e - = O .
ot
(23)
IMPACT OF MORTALITY ON PANJER'S MODEL
333
2. Uniform Random Censoring Assumption
Here it is assumed that for period j, ti has a uniform distribution on (a*,
~).
ffJ = JoI (b; - a])
=
ore- ~ ds dtj
e-op _ e-bl.
(b; -
~b?
1
~J = ,,z (b; - a;)
=_(
/'~ 1 a
(24)
a;) ,~
I~' lae_,~ ds dti
(e-°~ - e-b/~) /
(-b7 " a~-~ ]
(25)
q] = " 7 (b] - a])
"' [
(e -°*''~ _ e-Or)'~
=1 . . . . . .
,~ ~
(b; - a;) ,~
(26)
I
For each assumption, the partial derivatives of the probabilities can be found
and substituted into Equations (9) and (10) to yield the maximum likelihood
estimates of the parameters.
Unfortunately the data used by Partier and by Cowell and Hoskins cannot
be used to estimate the parameters in this model because there are no data
on the number of transitions directly to stage 6. It is hoped that the appropriate data will become available to the author (or to any other interested
party) for the purpose of estimating the parameters and testing the fit of this
model.
DISCUSSION OF PRECEDING PAPER
ERIC S. SEAI'I:
I congratulate the author for writing a fine paper on the important topic
of AIDS. In this discussion, I would like to briefly describe a model developed in the United Kingdom by Wilkie [1].
In this U.K. model, a life can be in any of the eleven states (see Figure
1). Note that the "Sick from AIDS" state in the U.K. model corresponds
to stages 2 to 4 combined in both Panjer's model and Ramsay's model (which
is an extension to Panjer's). Deaths during the "Positive" state (which
corresponds to stage 1, H I V + , in Panjer's and Ramsay's models) are provided for in the U.K. model. As pointed out by Ramsay, a shortcoming of
Panjer's model is that there is no such provision for an HIV+ life before a
full-blown AIDS is developed.
FIGURE 1
AIDS MODEL: STATES AND TRANSmONS
Immune,~4~ DeadfromImmune]
Positive ,
Dead
from
Clear
Dead
from
At risk
I
*
~',
~
Sick from AIDS
T
T,
Dead
from
Dead
from
Dead
from
Positive
Sick
AIDS
denotes possiblei n f e c t i o n .
One useful feature in the U.K. model is that the mortality rates are agespecific, and forces of progression to other states and of remaining in the
335
336
IMPACT OF MORTALITY ON PANTER'S MODEL
current state depend either on age or on both age and duration since entry
to the state. For example, the force of progression from the "Positive" state
to the "Sick from AIDS" state is a function of both age and the duration
since entry to the "Positive" state, and the mortality rate due to AIDS while
in the "Sick from AIDS" state is a function of age. Ramsay has noted in
the paper that "the assumption of constant forces in each stage may appear
to be unrealistic." It would be interesting to see how the Ramsay model can
be extended further to take age and duration into account.
REFERENCE
1. WILKIE,A. D. "An Actuarial Model for AIDS," Journal of the Institute of Actuaries
115 (1988): 839-53.
BEDA CHAN:
Dr. Ramsay has pointed out again [2] that one may die in stages 0, 1, 2,
or 3 and may die in stage 4 but with non-AIDS causes. While we accept
the WRSM reasoning that the stages are treated as physiologically sequential,
an individual may pass through stages so quickly relative to the frequency
of examination and observation that one is considered to have jumped stages.
We thus allow jumping to advanced stages as well as jumping directly to
the most advanced stage of death. To simplify our model, we do not allow
death from non-AIDS causes. Let
hazard rate from stage i to stage i + 1 + k
)@ ~.l,i =
where ~i is the hazard rate from stage i to i + 1. In other words, we assume
that the leaping rate is geometrically scaled down by the distance of the
leap, and for the sake of parsimony, we assume the leaping factor h is the
same for all stages. We consider the hazard matrix
-
0
0
0
0
0
O"
~o
kilo
0
~1
0
0
0
0
0
0
0
0
)g2~O
)~~'~l
~'L2
0
0
0
~z3
0
0
X3~Zo X2~z, X~2
--M~o X3Pq ~.2~z XP,3 P,4 O.
olscusstoN
337
where h is the only additional parameter. In particular, the hazard rates to
death in our model are:
h4~o = hazard rate from stage 0 to death
k31~ = hazard rate from stage 1 to death
h2~2 = hazard rate from stage 2 to death
X~3 = hazard rate from stage 3 to death
~4 = hazard rate from stage 4 to death,
which are comparable to that of Ramsay's [2, (54)]:
~/ = B c i f o r i = 0 , 1 , 2 , 3 .
Our model here allows for a geometric feature for the hazard rate to death,
like Ramsay, but also feature ~i as a factor that reflects the stability of stage
i. The current paper by Ramsay would have
" 0
~o
0
0
0
0
0
0
0
0-
0
~1
0
0
0
0
~2
0
0
0
0
W3
0
0
0
0
0
0
0
0
as its hazard matrix. The hazard matrix for the original Panjer paper is
-0
-
0
0
0
0
O-
Wo
0
0
0
0
g.~
0
0
0
0
0.2
0
0
0
0
g,3
0
0
0
0
0
0
0
0
0
0
0
0
Ix4
O-
338
IMPACT OF MORTALITYON FANTER)S MODEL
With the Frankfurt data, this hazard matrix is the best one can do. Our idea
of considering jumps towards advanced stages as well as to death is motivated by [1], is which Panjer considers distribution of the incubation period,
that is, movement through a number of stages.
We thank the author for pointing out the need for more detail in the data;
otherwise the deaths in stages 0, 1, 2, or 3 and deaths in stage 4 with nonAIDS causes cannot be quantitatively studied.
REFERENCES
1. PANJER, H. H. "AIDS: Some Aspects of Modeling the Insurance Risk," TSA XLI
(1989): 199-221.
2. RAMSAY,C. M. "AIDS and the Calculation of Life Insurance Functions," TSA XLI
(1989): 393--422.
(AUTHOR'S REVIEW OF DISCUSSION)
COLIN M. RAMSAY:
I thank Dr. Seah and Dr. Chan for participating in the discussion of my
paper.
Dr. Chan's model, which includes direct jumps to stages beyond the
immediate next stage, is an interesting extension of my model; it is worth
exploring further. One consequence of his model, however, is that lives will
develop AIDS faster. Interestingly, according to this model, a life that is in
stage 1 (that is, H I V + ) can instantaneously "die of full-blown AIDS" in
stage 5!
Dr. Seah's suggestions that age and duration should be included are well
taken. I am currently working on extending my model to include both age
and duration.